Extensions of orthosymmetric lattice bimorphisms

Let be an Archimedean vector lattice, let be its Dedekind completion and let be a Dedekind complete vector lattice. If is an orthosymmetric lattice bimorphism, then there exists a lattice bimorphism that not just extends but also has to be orthosymmetric. As an application, we prove the following: Let be an Archimedean -algebra. Then the multiplication in can be extended to a multiplication in , the Dedekind completion of , in such a fashion that is again a -algebra with respect to this extended multiplication. This gives a positive answer to the problem posed by C. B. Huijsmans in 1990.

     

Automatic continuity of -derivations on -algebras

Let be a -algebra acting on a Hilbert space , let be a linear mapping and let be a -derivation. Generalizing the celebrated theorem of Sakai, we prove that if is a continuous -mapping, then is automatically continuous. In addition, we show the converse is true in the sense that if is a continuous --derivation, then there exists a continuous linear mapping such that is a --derivation. The continuity of the so-called - -derivations is also discussed.

     

Poincaré duality algebras and rings of coinvariants

Let be a faithful representation of a finite group over the field . Via the group acts on and hence on the algebra of homogenous polynomial functions on the vector space . R. Kane (1994) formulated the following result based on the work of R. Steinberg (1964): If the field has characteristic 0, then is a Poincaré duality algebra if and only if is a pseudoreflection group. The purpose of this note is to extend this result to the case (i.e. the order of is relatively prime to the characteristic of ).

     

Almost simple groups of Suzuki type acting on polytopes

Let , with an odd power of two. For each almost simple group such that , we prove that is not a C-group and therefore is not the automorphism group of an abstract regular polytope. For , we show that there is always at least one abstract regular polytope such that . Moreover, if is an abstract regular polytope such that , then is a polyhedron.

     

Free modules of relative invariants and some rings of invariants that are Cohen-Macaulay

Let be a faithful representation of a finite group and a linear character. We study the module of -relative invariants. We prove a modular analogue of result of R. P. Stanley and V. Reiner in the case of nonmodular reflection groups to the effect that these modules are free on a single generator over the ring of invariants . This result is then applied to show that the ring of invariants for is Cohen-Macaulay. Since the Cohen-Macaulay property is not an issue in the nonmodular case (it is a consequence of a theorem of Eagon and Hochster), this would seem to be a new way to verify the Cohen-Macaulay property for modular rings of invariants. It is known that the Cohen-Macaulay property is inherited when passing from the ring of invariants of to that of a pointwise stabilizer of a subspace . In a similar vein, we introduce for a subspace the subgroup of elements of having as an eigenspace, and prove that Cohen-Macaulay implies is also.

     

A perturbed elementary operator and range-kernel orthogonality

Let denote the algebra of operators on a Hilbert . If and are commuting normal operators, and and are commuting quasi-nilpotents such that , then define and by , , and . It is proved that and , where is some scalar and is the quasi-nilpotent part of the operator .

     

Arens-Michael enveloping algebras and analytic smash products

Let be a finite-dimensional complex Lie algebra, and let be its universal enveloping algebra. We prove that if , the Arens-Michael envelope of is stably flat over (i.e., if the canonical homomorphism is a localization in the sense of Taylor (1972), then is solvable. To this end, given a cocommutative Hopf algebra and an -module algebra , we explicitly describe the Arens-Michael envelope of the smash product as an ``analytic smash product' of their completions w.r.t. certain families of seminorms.

     

On chaotic -semigroups and infinitely regular hypercyclic vectors

A -semigroup on a Banach space is called hypercyclic if there exists an element such that is dense in . is called chaotic if is hypercyclic and the set of its periodic vectors is dense in as well. We show that a spectral condition introduced by Desch, Schappacher and Webb requiring many eigenvectors of the generator which depend analytically on the eigenvalues not only implies the chaoticity of the semigroup but the chaoticity of every . Furthermore, we show that semigroups whose generators have compact resolvent are never chaotic. In a second part we prove the existence of hypercyclic vectors in for a hypercyclic semigroup , where is its generator.

     

The Banach algebra generated by a contraction

Let be a contraction on a Banach space and the Banach algebra generated by . Let be the unitary spectrum (i.e., the intersection of with the unit circle) of . We prove the following theorem of Katznelson-Tzafriri type: If is at most countable, then the Gelfand transform of vanishes on if and only if

     

The rank of elliptic curves with rational 2-torsion points over large fields

Let be a number field, an algebraic closure of , the absolute Galois group , the maximal abelian extension of and an elliptic curve defined over . In this paper, we prove that if all 2-torsion points of are -rational, then for each , has infinite rank, and hence has infinite rank.

     

Simple real rank zero algebras with locally Hausdorff spectrum

Let be a unital, simple, separable -algebra with real rank zero, stable rank one, and weakly unperforated ordered group. Suppose, also, that can be locally approximated by type I algebras with Hausdorff spectrum and bounded irreducible representations (the bound being dependent on the local approximating algebra). Then is tracially approximately finite dimensional (i.e., has tracial rank zero).

Hence, is an -algebra with bounded dimension growth and is determined by -theoretic invariants.

The above result also gives the first proof for the locally case.

     

A Beurling-Carleson set which is a uniqueness set for a given weighted space of analytic functions

Let be a sequence of positive real numbers. We define as the space of functions which are analytic in the unit disc , continuous on and such that

where is the Fourier coefficient of the restriction of to the unit circle . Let be a closed subset of . We say that is a Beurling-Carleson set if

where denotes the distance between and . In 1980, A. Atzmon asked whether there exists a sequence of positive real numbers such that for all and that has the following property: for every Beurling-Carleson set , there exists a non-zero function in that vanishes on . In this note, we give a negative answer to this question.

     

The real rank zero property of crossed product

Let be a unital -algebra, and let be a -dynamical system with abelian and discrete. In this paper, we introduce the continuous affine map from the trace state space of the crossed product to the -invariant trace state space of . If is of real rank zero and is connected, we have proved that is homeomorphic. Conversely, if is homeomorphic, we also get some properties and real rank zero characterization of . In particular, in that case, is of real rank zero if and only if each unitary element in with the form can be approximated by the unitary elements in with finite spectrum, where , , and if moreover is a unital inductive limit of the direct sums of non-elementary simple -algebras of real rank zero, then the above can be cancelled.

     

Traces and Sobolev extension domains

Assume that is a bounded domain and its boundary is -regular, . We show that if there exists a bounded trace operator , and , and -Hölder continuous functions are dense in , , then the domain is a -extension domain.

     

Lightface -indescribable cardinals

-absoluteness for forcing means that for any forcing , . `` inaccessible to reals' means that for any real , . To measure the exact consistency strength of `` -absoluteness for forcing and is inaccessible to reals', we introduce a weak version of a weakly compact cardinal, namely, a (lightface) -indescribable cardinal; has this property exactly if it is inaccessible and .

     

On Burch's inequality and a reduction system of a filtration

Let be a multiplicative filtration of a local ring such that the Rees algebra is Noetherian. We recall Burch's inequality for and give an upper bound of the a-invariant of the associated graded ring using a reduction system of . Applying those results, we study the symbolic Rees algebra of certain ideals of dimension .

     

On operators which commute with analytic Toeplitz operators modulo the finite rank operators

It is shown that an operator on the Hardy space (or ) commutes with all analytic Toeplitz operators modulo the finite rank operators if and only if . Here is a finite rank operator, and in the case , is a sum of a rational function and a bounded analytic function, and in the case , is a bounded analytic function.

     

Resolutions of ideals of fat points with support in a hyperplane

Let be a fat point subscheme of , and let be a linear form such that some power of vanishes on (i.e., the support of lies in the hyperplane defined by , regarded as ). Let , where is the subscheme of residual to ; note that is a fat points subscheme of . In this paper we give a graded free resolution of the ideal over , in terms of the graded minimal free resolutions of the ideals . We also give a criterion for when the resolution is minimal, and we show that this criterion always holds if .

     

A theorem on reflexive large rank operator spaces

If every nonzero operator in an -dimensional operator space has rank , then is reflexive.

     

Isomorphic -subspaces in Orlicz-Lorentz sequence spaces

Given a decreasing weight and an Orlicz function satisfying the -condition at zero, we show that the Orlicz-Lorentz sequence space contains an -isomorphic copy of , if and only if the Orlicz sequence space does, that is, if , where and are the Matuszewska-Orlicz lower and upper indices of , respectively. If does not satisfy the -condition, then a similar result holds true for order continuous subspaces and of and , respectively.